Theorem csbima12 5344

Description: Move class substitution in and out of the image of a function. (Contributed by FL, 15-Dec-2006.) (Revised by NM, 20-Aug-2018.)

Assertion

Ref	Expression
csbima12	|- [_ A / x ]_ ( F " B ) = ( [_ A / x ]_ F " [_ A / x ]_ B )

Proof of Theorem csbima12

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbeq1 3423	. . . 4 |- ( y = A -> [_ y / x ]_ ( F " B ) = [_ A / x ]_ ( F " B ) )
2		csbeq1 3423	. . . . 5 |- ( y = A -> [_ y / x ]_ F = [_ A / x ]_ F )
3		csbeq1 3423	. . . . 5 |- ( y = A -> [_ y / x ]_ B = [_ A / x ]_ B )
4	2, 3	imaeq12d 5328	. . . 4 |- ( y = A -> ( [_ y / x ]_ F " [_ y / x ]_ B ) = ( [_ A / x ]_ F " [_ A / x ]_ B ) )
5	1, 4	eqeq12d 2465	. . 3 |- ( y = A -> ( [_ y / x ]_ ( F " B ) = ( [_ y / x ]_ F " [_ y / x ]_ B ) <-> [_ A / x ]_ ( F " B ) = ( [_ A / x ]_ F " [_ A / x ]_ B ) ) )
6		vex 3098	. . . 4 |- y e. _V
7		nfcsb1v 3436	. . . . 5 |- F/_ x [_ y / x ]_ F
8		nfcsb1v 3436	. . . . 5 |- F/_ x [_ y / x ]_ B
9	7, 8	nfima 5335	. . . 4 |- F/_ x ( [_ y / x ]_ F " [_ y / x ]_ B )
10		csbeq1a 3429	. . . . 5 |- ( x = y -> F = [_ y / x ]_ F )
11		csbeq1a 3429	. . . . 5 |- ( x = y -> B = [_ y / x ]_ B )
12	10, 11	imaeq12d 5328	. . . 4 |- ( x = y -> ( F " B ) = ( [_ y / x ]_ F " [_ y / x ]_ B ) )
13	6, 9, 12	csbief 3445	. . 3 |- [_ y / x ]_ ( F " B ) = ( [_ y / x ]_ F " [_ y / x ]_ B )
14	5, 13	vtoclg 3153	. 2 |- ( A e. _V -> [_ A / x ]_ ( F " B ) = ( [_ A / x ]_ F " [_ A / x ]_ B ) )
15		csbprc 3807	. . 3 |- ( -. A e. _V -> [_ A / x ]_ ( F " B ) = (/) )
16		csbprc 3807	. . . . 5 |- ( -. A e. _V -> [_ A / x ]_ B = (/) )
17	16	imaeq2d 5327	. . . 4 |- ( -. A e. _V -> ( [_ A / x ]_ F " [_ A / x ]_ B ) = ( [_ A / x ]_ F " (/) ) )
18		ima0 5342	. . . 4 |- ( [_ A / x ]_ F " (/) ) = (/)
19	17, 18	syl6req 2501	. . 3 |- ( -. A e. _V -> (/) = ( [_ A / x ]_ F " [_ A / x ]_ B ) )
20	15, 19	eqtrd 2484	. 2 |- ( -. A e. _V -> [_ A / x ]_ ( F " B ) = ( [_ A / x ]_ F " [_ A / x ]_ B ) )
21	14, 20	pm2.61i 164	1 |- [_ A / x ]_ ( F " B ) = ( [_ A / x ]_ F " [_ A / x ]_ B )

Syntax hints:   -. wn 3   = wceq 1383   e. wcel 1804   _Vcvv 3095   [_csb 3420   (/)c0 3770   "cima 4992

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-br 4438  df-opab 4496  df-xp 4995  df-cnv 4997  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002

This theorem is referenced by:  csbrn  5458  disjpreima  27421